Pointwise convergence of polynomial multiple ergodic averages along the primes
Renhui Wan
TL;DR
The paper proves pointwise almost everywhere convergence for the prime-weighted multilinear polynomial ergodic averages A_{N,Λ;X}^{P_1,...,P_k} when the polynomials P_i have integer coefficients and pairwise distinct degrees. It develops a multilinear circle method tailored to von Mangoldt weights, combining inverse theorems, a multilinear Weyl inequality, a multilinear Rademacher–Menshov inequality, and an arithmetic multilinear estimate to control minor and major arc contributions. The Calderón transference principle reduces the problem to the integer shift setting, where the authors implement a sophisticated adelic harmonic-analysis framework and Ionescu–Wainger multiplier theory to handle the discrete-continuous interplay. As a consequence, they establish pointwise convergence and a lacunary-variation bound, extending prior norm-convergence results (Wooley–Ziegler) to the prime-weighted, multilinear, distinct-degree setting and enabling a broader understanding of prime-driven polynomial ergodic averages. The work introduces principal innovations in inverse theory for Cramér-weighted averages, multilinear Weyl-type control, and arithmetic multilinear estimates, with potential impact on ergodic theory and additive combinatorics through refined major/minor arc analysis and adelic methods.
Abstract
We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages $$\frac{1}{N} \sum_{n=1}^N \La(n) f_1(T^{P_1(n)} x)\cdots f_k(T^{P_k(n)} x)$$ as $N\to \infty$, where $\La$ is the von Mangoldt function, $T \colon X \to X$ is an invertible measure-preserving transformation of a probability space $(X,ν)$, $P_1,\ldots, P_k$ are polynomials with integer coefficients and distinct degrees, and $f_1,\ldots,f_k\in L^\infty(X)$. This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. Building on the foundational work of Krause--Mirek--Tao (Ann. of Math., 2022), Kosz--Mirek--Peluse--Wright (arXiv: 2411.09478, 2024), and Krause--Mousavi--Tao--Teräväinen (arXiv: 2409.10510, 2024), we develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages. This method combines harmonic analysis techniques across multiple groups with the newest inverse theorem from additive combinatorics. In particular, the principal innovations of this framework include: (i) an inverse theorem and a Weyl-type inequality for multilinear Cramér-weighted averages; (ii) a multilinear Rademacher-Menshov inequality; and (iii) an arithmetic multilinear estimate.